Related papers: Factorization and root-finding for polynomials ove…
We study differential splitting fields of quaternion algebras with derivations. A quaternion algebra over a field $k$ is always split by a quadratic extension of $k$. However, a differential quaternion algebra need not be split over any…
We describe several algorithms for computing $e$-th roots of elements in a number field $K$, where $e$ is an odd prime-power integer. In particular we generalize Couveignes' and Thom\'e's algorithms originally designed to compute…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
Many matrices associated with fast transforms posess a certain low-rank property characterized by the existence of several block partitionings of the matrix, where each block is of low rank. Provided that these partitionings are known,…
It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous $G$-algebras, are finite factorization domains (FFD for short). Utilizing this result, we contribute an algorithm to…
We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…
This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…
In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient…
Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for…
Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…
Let $f(x)$ be a separable polynomial over a local field. Montes algorithm computes certain approximations to the different irreducible factors of $f(x)$, with strong arithmetic properties. In this paper we develop an algorithm to improve…
We present a FFT-based algorithm for the computation of a polynomial's coefficients from its roots, and apply it to obtain the coefficients of interpolation polynomials, to invert Vandermondians and to evaluate the symmetric functions of a…
In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the…
Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate…
This paper presents new formulary solutions for quantic polynomial equations in general forms, where we present five solutions for any fifth degree polynomial equation with real coefficients, and thereby having the possibility to calculate…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
Expectation propagation is a general approach to fast approximate inference for graphical models. The existing literature treats models separately when it comes to deriving and coding expectation propagation inference algorithms. This comes…
Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the…
This paper presents an experimental study on the application of quaternions in several machine learning algorithms. Quaternion is a mathematical representation of rotation in three-dimensional space, which can be used to represent complex…
We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…